We theoretically investigate coupled Fano structures which combine the characteristics of both directly coupled Fabry-Perot cavities (DCFPC) and a side-coupled integrated spaced sequence of resonators (SCISSOR). Asymmetric and symmetric Fano resonances in a single and doubly-coupled Fano unit are analytically derived based on Fabry-Perot approach. It is found that doubly-coupled Fano units show a special asymmetric EIT-like lineshape. This structure shows an index changing sensitivity of 10−6, about two orders higher than that of the single Fano resonator, which is promising for index sensor application. A unique frequency detuning method of EIT like lineshape is also found in the doubly-coupled Fano units. The multiple coupled Fano unit structure demonstrates potentials for applications of tunable optical filter and slow light, whereas for the latter it shows much higher group delay than that of SCISSORS and DCFPC with the same parameters.
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The simplest resonator system comprising an optical waveguide directly or indirectly coupled with an optical cavity normally exhibits symmetric resonance lineshapes in the transmission spectrum. The realization of such a resonator structure can be done using various platforms, i.e., photonic crystal (PC) or whispering gallery resonators such as microring resonators. On a PC platform, the resonator system is constructed by defining a ‘point defect’ as a resonator and a ‘line defect’ as an optical waveguide in an otherwise periodic media. On the other hand, a whispering gallery resonator is a close-loop waveguide which is functionally similar to a Fabry-Perot (FP) etalon, where the evanescent field coupling mimics the function of mirrors in an FP cavity. Using such resonant properties, various applications have been proposed and demonstrated, including all-optical switches, modulators, and channel-dropping filters [1–4].
Recently, research interests have shifted to a system consisting of only two resonators interacting in unique ways. For example, the optical analogue of Fano resonance that generally exhibits asymmetric resonance lineshape has been theoretically explored for reducing switching threshold power , and improving optical switching characteristics [6–8]. A more specific and notable example would be the optical analogue of electromagnetically induced transparency (EIT), where a near unity transmission is produced in the absorption spectrum by means of destructive interference of decaying cavity fields [9,10]. Although these Fano resonator systems have been widely explored from whispering gallery resonators to photonic crystals [5, 11], the periodic cascade of such structures, to the best of our knowledge, has not been investigated thus far.
In this paper, we theoretically study the transmission and photonic band characteristics of periodic cascade of Fano resonators [Fig. 1(c) ] with the following outline. In Section 2, we start by analytically deriving the property of a single Fano unit by using Fabry-Perot approach. In Section 3, we briefly analyze the transmission characteristics of two coupled-Fano units, where sharp Fano resonance as a function of cavity detuning and mirror reflectivity is investigated. In this part, application on the index sensor is also discussed. In Section 4, we extend this approach to transfer matrix formalism in dealing with coupled Fano units. Schematically, the proposed configuration resembles the combination of directly coupled cavities [12, 13, 28] (in ring resonator platform, this is often addressed as CROW) and side-coupled integrated spaced sequence of resonator (SCISSOR) [14–16] as shown in Fig. 1(a) and Fig. 1(b), respectively.
In this paper, a particular attention is focused on directly coupled Fabry-Perot cavities (DCFPC), where each of FP cavities is constructed by two partially reflecting elements formed by insertion of holes in PhC waveguide  or a slight lateral offset of waveguide . Comparison between coupled Fano units, SCISSOR, and DCFPC is also drawn and discussed. One of our motivations in this paper is to extend the analysis where many of these two-cavity systems are coupled in one dimensional array. It is found that the interference between two resonant pathways in each two-cavity system evolves to the interaction between two continuous bands, which are originated from DCFPC and SCISSOR-like periodic structures in the coupled-cavity configuration. Applications of periodic cascade of Fano units on tunable optical filter and optical delay line are briefly discussed. Finally, we summarize and conclude our findings in the last section.
2. Single Fano unit
The Fano resonator unit considered in this paper consists of one side-coupled cavity and one Fabry-Perot cavity formed by two partially reflecting elements [see Fig. 2(a) ]. The side-coupled cavity considered in this configuration is a standing wave resonator. The simple mirror model [Fig. 2(b)&(c)] is considered in the analysis, where the insignificant spectral dependence of reflectivity and transmission of those elements is ignored. This is because the spectrum of interest for theoretical calculation that concerns the high-Q regime of resonators is much narrower than the span of the spectral variation. The simplified model given in Fig. 2(b)&(c) is valid for the weak coupling regime and thus the whole analysis is limited to that as well. As the side-coupled cavity has been shown to exhibit reflection and transmission , the FRU can be decomposed into two directly coupled FP cavities, where the first is formed between the first mirror (M 1) and the side-coupled cavity (C), and the second is between the cavity C and the second mirror (M 2). Thus, using the property of Fabry-Perot transmission, it is possible to deduce the transmission characteristics of FRU. The structure can be simplified if the first FP cavity is reduced into an equivalent frequency dependent mirror. The equivalent mirror and the second mirror form an effective FP cavity with half the total cavity length, as shown in Fig. 2(c). The lossless transmittance (t s) and reflectance (r s) of a side-coupled cavity are ,Eq. (2). When the perturbation from the side-coupled cavity is hardly felt in the waveguide [(ω-ω 0) >> γc, t s~1, r s~0], one can have the background spectrum identical to that in FP cavity,Eq. (2) is reduced to
The denominator of Eq. (2) shows a mixture of resonant process originated from the side-coupled resonator and the FP cavity. The two cavity modes couples by way of destructive interference, resulting in a sharp dip in the vicinity of the side-coupled cavity resonance (ω = ω 0). Figure 3(a) shows the plots for lossless transmissions under different cavity resonance detuning in a fixed resonance broadening and mirror reflectivity, which are chosen to be γc = 2x10−3(2πc/L) and r p = 0.4, respectively. The FP resonance frequency (for m = 1) is fixed to 0.25(2πc/L), while the side-coupled cavity resonance varies from 0.2(2πc/L) to 0.3(2πc/L). In the case where the cavities are co-resonant (ω 0 = ω FP), one would have the situation of the spectrum following that of the FP response (as indicated in dashed line), with a symmetric dip at ω = ω FP = ω 0. The symmetric split-resonance corresponds to the equal electric field localization in both the side-coupled and the FP cavity. When the cavities are not co-resonant (ω 0≠ω FP), one has the situation of asymmetric and sharp resonance enveloped by the FP spectrum. The degree of lineshape asymmetry generated by this processes depends on the cavity resonance detuning (Δω 12 = ω 0-ω FP) as well as the resonance broadening of both cavities. The field distribution is the same as other two-cavity structures [6,18], where each of the asymmetric split-resonance corresponds to a dominant light localization in the cavity whose resonance frequency is in the close proximity. At the broader split-resonance, the field dominates in the FP cavity. At narrow and asymmetric resonance, the field dominates in the side-coupled cavity.
Naturally, the lineshape becomes the sharpest and the most asymmetric when the cavity resonance detuning is at the farthest, which is half the free-spectral-range (FSR). As indicated in Eq. (4), the property of the Fano resonance is dominated by that of the side-coupled cavity. This means Fano resonance only occurs when the criteria of intra-cavity intensity buildup in the side-coupled cavity is satisfied, i.e., over-coupling condition (γc>γ0) . In under-coupling condition (γc<γ0), there is only one cavity mode involved in the process because there is no sufficient intensity buildup in the side-coupled cavity. Thus, the Fano resonance cannot be formed under this condition. This is shown in Fig. 3(b), where the transmission of the most asymmetric and sharpest resonance [i.e., ω 0 = 0.375(2πc/L)] is plotted for the same γc under difference losses (γ0). In the lossless condition (γ0 = 0), the contrast of the transmission is unity. In the over-coupling situation under the presence of loss, i.e., γ0 = 10−3(2πc/L), the Fano resonance still exists but with decreased contrast. In the under-coupling condition, i.e., γ0 = 4x10−3(2πc/L), the Fano resonance no longer exists and the transmission approximates the FP spectrum. Similar phase sensitivity enhancement effects have previously been observed in other filters such as a Fabry-Perot etalon microtoroid resonator coupled system .
3. Doubly-coupled Fano units
In the case where the resonator units are cascaded, normally symmetric resonance splitting arises with the amount of peaks dependent on the number of units involved [12,14]. There are two ways of cascading Fano units. The first and more general configuration is shown in Fig. 4(a) , where each unit has two mirrors and is separated by a distance d between each other. The special configuration is shown in Fig. 4(b), where one Fano unit share a common reflector with another. We begin with the simplest case where only two units are coupled. Following the FP approach from the previous section, the transmission of two Fano units separated by distance d isEq. (5). Here, we show two ways of tuning Fano resonance. Figure 5 shows the lossless transmission of two Fano units in both special and general case, as a function of mirror reflectivity. The spectrum of single Fano unit is indicated by the dash lines, while the ω 0 and ω FP are fixed to 0.15(2πc/L) and 0.25(2πc/L), respectively.
As shown in Fig. 5(a), the Fano resonance splits into two asymmetric resonances, following the spectrum of single Fano unit. The split-resonance is broader (narrower) when it approaches the maximum (minimum) side of the single Fano unit spectrum. It can be seen that the resonance splitting is insensitive to the change of r p for the special case. In the general configuration, however, one has different situation, where the linewidth can be significantly reduced by adjusting r p. This is shown in Fig. 5(b), where d = L is chosen as example. The FP cavity formed in between the two Fano units helps to facilitate the coupling, and thus increases the sensitivity towards the change in the resonance condition, i.e., arg[r u1 r u2exp(i2θ)] = m2π. This can be done by changing γc, ω 0, r p, and d. The second way of tuning Fano resonance is illustrated in Fig. 6 , where ω 0 is tuned around anti-resonance of FP cavity, i.e., |(ω 0-ω FP)/ω FP| = 0.5, for a fixed reflectivity. The general configuration clearly exhibits a wider tunability compared to that in the special one for a fixed range of cavity resonance detuning. Towards the FP anti-resonance, i.e., at ω 0 = 0.13(2πc/L), it can be seen that the Fano linewidth becomes extremely small. This mechanism is to be contrasted with the classical EIT case. In classical EIT structure, the two coupled cavities are two resonances with symmetric Lorentzian lineshape, while the resonance linewidth is inversely related to the cavity resonance detuning of two side-coupled cavities . In our structure, the corresponding resonance is replaced by Fano resonator, and as a result, the asymmetric EIT structure is formed on the output transmission spectrum. In the asymmetric EIT sturucture, the resonance linewidth is inversely related to the detuning of ω0 relative to the anti-resonance of FP cavity.
Fano lineshape has many advantages compared with general cavity with symmetric Lorentzian lineshape. As mentioned in reference , one of the most interest applications of Fano resonance lies on the index sensing. It can test a 10−4 index changing in a point defect photonic crystal slab with Q exceeding 15000. It should be noted that even higher sensitivity in doubly-coupled Fano units can be realized than that in the single Fano resonator. For example in Fig. 5(b), when rp = 0.2 which can be easily realized in photonic crystal by adding two air holes along the line defect waveguide, the transmission increases from 0 to 99% with a normalized detuning of (ω-ω FP)/ω FP = 10−4 which means, for the side coupled cavity with same Q factor, the sensitivity of index changing reaches 10−6 which is two orders higher than that of the single Fano resonator. Tuning of the working frequency in general EIT structure is usually realized by changing the separation distance between two cavities . In doubly-coupled Fano units, this can be done according to r p as shown in Fig. 5. Because of the FP cavity effect formed by the two partial reflectors between the two side-coupled cavities, a shift of r p will induce the change in the optical feedback strength between the two cavities. As a result, tuning the working frequency of the index sensor through r p in doubly-coupled Fano units offers a unique way in contrast to that in a general EIT structure. For example, in general EIT structure as demonstrated in ref , by changing d from d = L to d = 1.2L, the normalized detuning of the working frequency is −0.0026. The same extent of detuning can be realized in doubly-coupled Fano units by fixing d = L and changing r p from r p = 0.4 to r p = 0.5. This unique way of frequency detuning may be useful for optical modulation and switching. For example, in silicon photonic crystal doubly-coupled Fano units, the partial reflectors can be one or two air holes along the PC line defect waveguide. For the case where it is unable to offer sufficient index change in the cavity, the output can be modulated via r p tuning. Holes filled with nonlinear polymer or other active material [20,21] can be adopted for r p tuning when the index of the filled material is changed via electrical or optical means. It is seen that this kind of optical switching needs only small shift of r p, whereas further discussion is not within the scope of this paper.
One effect of fabrication error on our configuration, most likely due to dose variation among cavity structures, is the change of the hole-diameter of the inter-cavity Bragg mirrors, which results in the change of reflectivity. The increase (decrease) in mirror reflectivity gives result in stronger (weaker) optical feedback, resulting in the change of roll-off in the flat-band for flat-top filter application and the change in the resonance finesse of narrow resonance in doubly coupled Fano resonance. The impact of cavity size deviation can be minimized when the cavity size is maintained to be small, which translates to much smaller resonance order compared to that in ring resonators. However, the effect of cavity size detuning in photonic crystal based cavities is much less serious than that in ring resonators.
4. Multiple coupled Fano units
In dealing with multiple cascades of Fano units, it is necessary to transform the Fabry-Perot expressions described in Section 2 into transfer matrices. Generally, the transfer matrix of an element characterized by reflection (r q) and transmission (t q) can be derived as
It should be noted that the cascade of Fano units resembles the combination of DCFPC and SCISSOR structures . The structure reduces to DCFPC when γc = 0, in the form of directly coupled FP cavities, with each FP having a cavity length of 2L. When r P = 0, the structure reduces to SCISSOR in the form of evenly spaced side-coupled resonators. The spectrum of DCFPC, SCISSOR, and coupled Fano units are shown in Fig. 7 . Here, we assume that the Fano units are cascaded in a special configuration (N = 20). Before explaining the characteristics of coupled Fano units, it is necessary to briefly describe the property of DCFPC and SCISSOR.
The DCFPC shown in Fig. 1(a) consists of a series of coupled FP resonators. The light propagates by means of mode-hopping between neighboring FP cavities through direct coupling . This causes the formation of photonic band centered at FP resonance in the form of N-fold resonance splitting, where N is the number of units in the cascade. The photonic bandgap is centered at the FP anti-resonance and formed due to the presence of optical feedback in between the cavities, which is controlled by the mirror reflectivity r P. The higher the reflectivity, the wider is the photonic bandgap. The density of the ripples is indicative of the group index of the propagating light. Near the band-edge, the ripple is getting sharper, indicating that the light propagation is greatly slowed down and extremely dispersive.
In the SCISSOR structure, as shown in the middle plot, the cavity modes result from interaction by two mechanisms. The more dominant one is the summation of decaying fields from each cavity. By this mechanism, a full photonic band, i.e., characterized by a zero transmission, is formed near ω = ω 0. This type of bandgap is denoted as resonator-gap [14, 24]. The second mechanism is the Bragg interference between cavity modes, in which a photonic bandgap is formed and centered at ω = ω FP. This type of bandgap is denoted as the Bragg gap. The Bragg gap becomes stronger when the structure has more unit cells, as is usually the case for any Bragg gratings.
Generally, as shown in Fig. 7(a), the two mechanisms do not interact and normally result in two distinct photonic bandgaps. It should be noted that the ripples are rather chirped in between two bandgaps, where the ripples are much denser near the resonator-gap. In this situation, the propagating light experiences a mixture of the two mechanisms, where the spectrum of light near the resonator-gap will be much more dispersive compared to that near the Bragg-gap. When the two mechanisms occur at the same frequency, the optical feedback provided by the Bragg-gap is resonantly amplified by the cavities. Thus, this makes the faster development of photonic bandgap in a smaller number of unit cells. This is shown in Fig. 7(b), where a flat-top transmission with broad gap-width is formed with very few ripples. Furthermore, the bandgap is formed in a much smaller number of unit cells, compared to that in conventional Bragg gratings.
When the above two array structures are superimposed, we can have the situation of coupled Fano units shown in the lower plot of Fig. 7. A mixture of the DCFPC and SCISSOR characteristics is clearly manifested, where a reflection band is formed within the broader transmission band. The direct coupling of Fano units, as in the case of DCFPC, gives rise to multiple Fano resonance splitting which forms a Fano reflection band, centered at ω 0. It can be seen that a spectrum of a single Fano unit directly translated to the formation of Fano band, with the ripples asymmetrically chirped. The mechanism of Fano-band formation can also be seen from the photonic band structure perspectively. The light propagation in the infinitely long periodic structure for the special case can be well described by using Bloch theorem,
Figure 8 presents the photonic bands of DCFPC (CB), SCISSOR (SB), and coupled Fano units (FB), where ω 0 = ω FP and the bandgaps are indicated in the shaded area. The resonant pathway in a single resonance unit directly translates into a photonic band, corresponding to Bloch modes in the band structure. The interaction of DCFPC bands and SCISSOR bandgap is clearly manifested, where a split of band is formed around resonance (ω = ω 0 = ω FP), with the properties asymptotically converge to those of DCFPC (far from bandgap) and SCISSOR (near the bandgap). The bandgap formed in between the two split-bands is the Fano bandgap shown earlier (see Fig. 7), where the gap width is comparable with the degree of optical feedback in between side-coupled cavities, as controlled by γc. It can be seen that the band is nearly flat near the Fano band-edge over certain frequency band. This is not the case in the conventional photonic band structure, where such a flat band only occurs in a very small band of frequency near the band-edge. This may be useful in slow light structure, where a large group index n g (which corresponds to the flat band) is desired over a frequency band.
It should be noted that such interaction does not exists between DCFPC gap and SCISSOR band around the anti-resonance [|(ω-ω FP)/ω FP| = 0.5]. This can be explained by the nature of Fano resonance that involves direct and indirect optical pathways. In the situation where DCFPC gap overlaps with SCISSOR band there is only one resonant optical pathway, which is the mode-hopping between FP cavities formed by the mirrors. The other pathway is the non-resonant light propagation along the straight waveguide. Thus, there is no interference between FP and side-coupled cavities, and thus the DCFPC characteristic dominates. This is in contrast with Ref , in which the SCISSOR characteristics dominate in two-dimensional resonator arrays, due to the fact that the optical pathways of SCISSOR and DCFPC structure are defined in different directions.
The case where the FP and the side-coupled cavities are not co-resonant is also shown in Fig. 9 , where the band structure for different ω 0 is shown side-by-side with the reflection spectrum. The way the two bands interact still follows the same mechanism as that in co-resonant case. Since there are two distinct bandgaps in SCISSOR when ω 0≠ω FP, there is a situation in which the Bragg gap overlaps with the DCFPC continuous band. It can be shown that the two bands do not interact in the same way since the DCFPC gap does not interact with the SCISSOR band at anti-resonance for coresonant case (see Fig. 8). The only difference is the shift of Fano band with ever increasing asymmetricity in the ripples as it moves farther from the SCISSOR gap. Such a shift in Fano bands may find application in tunable band-stop filter. We have shown that the Fano band can be nearly flat over a certain frequency band and that the Fano band is the direct translation of the single Fano unit’s properties. In addition, tuning ω 0 away from ω FP gives more bandwidth to the flat-band (see Fig. 9). The farther is the center of Fano band from the SCISSOR bandgap, the more bandwidth is likely to be acquired for a large group index. From the perspective of single Fano unit, this translates to ever increasing Q-factor of the Fano resonance, as it moves towards to the FP anti-resonance. It is shown in Fig. 5 and Fig. 6 that high-Q Fano resonance is more easily obtained in general configuration, because of the more sensitive resonance condition due to the existence of FP cavity in between Fano units. Thus, coupling of Fano units (chosen for highest Q-factor) in general cascade configuration will give a very flat photonic band with the optimum bandwidth. In Fig. 10 , we present the group delay for DCFPC, SCISSORS and general cascade configurations, side by side with their corresponding transmission spectra.
Here, ω 0 = 0.13(2πc/L) is chosen as an example of the highest achievable Q-factor in general cascade configuration. From Fig. 10, it can be seen that the group delay for the general cascade is almost 300 times higher than that of the DCFPC and 4 times higher than that of SCISSORS. However, the corresponding transmission for the general cascade is very much densely rippled compared to the special counterparts. In addition, the transmission is very sensitive to loss. Simulation has shown that to maintain insertion loss of 30dB, the maximum permissible loss is around γ0 = 10−5(2πc/L). This suggests the intrinsic Q-factor of the cavity should be at least 100 times more than the external Q, suggesting an intrinsic Q factor on the order of 105-106, which is relatively possible in recently fabricated photonic crystal cavities [25, 26]. From the perspective of optical buffer, the ripple induces large signal distortion which limits the usable bandwidth. In fact, this is the familiar problem in implementing coupled resonator optical waveguide (CROW) as the optical buffer . The ripples can be reduced by adiabatically change the mirror reflectivity from very small (at the edge of the devices) to the desired values (at the center of the devices). This is a well known apodization technique in conventional filter synthesis [28, 29].
5. Summary and conclusions
Coupled-Fano structures have been theoretically investigated. It is shown that a unique asymmetric EIT-like lineshape is formed in doubly-coupled Fano units. It is found that for the application of index sensing, a 10−6 index changing can be detected using this structure. Moreover, a unique frequency detuning method of the EIT is found in doubly-coupled Fano structure which is to tune the frequency of EIT by changing of rp. Based on this method, doubly coupled Fano units may also be utilized on optical modulation and switching. For the multiple coupled Fano units, it has been shown that Fano band is formed by virtue of interaction of SCISSOR and DCFPC bands. Tuning of ω 0 can be used to shift the Fano bandgap, suggesting the possibility of tunable flat-top filter application. Finally, we compare the DCFPC, SCISSORS and cascade Fano units on the application of slow light. The flattening of Fano band results in higher group delay near the bandgap of Fano unit than other structures which can be utilized on optical delay line.
X. Tu and L. Y. Mario made equal contribution in this paper. This work was supported by the Singapore National Research Foundation under CRP Award No. NRF-G-CRP 2007-01and was also supported in part by the Ministry of Education (MOE), Singapore (Research Grant No. ARC 16/07). T. Mei acknowledges support from Department of Education of Guangdong Province, China (Grant No. C10131).
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